University of Illinois Urbana-Champaign

On lower bounds for the betti numbers of finite length modules

Charalambous, Hara

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Description

Title
On lower bounds for the betti numbers of finite length modules
Author(s)
Charalambous, Hara
Issue Date
1990
Doctoral Committee Chair(s)
Griffith, Phillip A.
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
Abstract
  • In this manuscript we consider multigraded modules. Chapter 1 gives the necessary definitions and examples that develop the theory of multigraded modules. A multigraded module has a multigraded minimal resolution. We give the necessary conditions for a matrix to correspond to a multigraded map and show that all associated primes of a multigraded module are multigraded.
  • In Chapter 2 we show that the betti numbers of multigraded modules satisfy the bound: $\beta\sbsp{i}{R}(M) \geq (\sbsp{i}{d})$. The multigraded modules that are not isomorphic to R modulo an R-sequence can be divided into two categories according to their betti numbers. Either for all i $\beta\sbsp{i}{R}(M) \geq (\sbsp{i}{d}) + (\sbsp{i-1}{d-1})$ or for all i $\beta\sbsp{i}{R}(M) \geq (\sbsp{i}{d}) + (\sbsp{\ i}{d-1})$. Thus if $\beta\sbsp{i}{R}(M)$ = $(\sbsp{i}{d})$ then M must be R modulo a maximal R-sequence. In case M is an almost complete intersection we derive its betti numbers.
  • In Chapter 3 we give better bounds for the betti numbers of cyclic multigraded modules using a deformation argument combined with localization. Lastly we point out that the sum of the betti numbers for all modules of finite length is greater than or equal to 2$\sp{d}$ + 2$\sp{d-1}$ where d is the dimension of the ring up to dimension 4.
Type of Resource
text
Permalink
http://hdl.handle.net/2142/19766
Copyright and License Information
Copyright 1990 Charalambous, Hara

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